from an unequally heated space. 103 



rature to which the body is brought by this process of restora- 

 tion ; let t be the temperature of the body in its given condition 

 at any point xyz ; and let cdt . dx dy dz be the quantity of heat 

 that an infinitely small element^ dx dy dz, of the body at this 

 point must part with to go down in temperature from t to t—dt. 

 Let us suppose that this quantity of heat enters a perfect thermo- 

 dynamic engine, of which the hot part is at the temperature t, 

 and the cold part at the temperature T. The quantity of work 

 that will be derived from it will be 



dxdydzZ.cdtJ:\-e^f'^^^'] 



... («), 



where [i denotes Carnot's function, and J the mechanical equi- 

 valent of the thermal unit (see Dynamical Theory of Heat, § 25, 

 Trans. Royal Soc. Edinb. 1851, or Phil. Mag. Aug. 1852), and 

 the part of it rejected as waste into the refrigerator at the tem- 

 perature T will be 



dxdydz.cdt.e^^'^^ {b), 



or dxdy dz .cdt .€^^ ^ , . 



--1 /"T 



y-,,> 



Hence the whole work obtained by lowering, by means of a per- 

 fect thermo-dynamic engine, the temperature of the element 

 dx dy dz from ^ to T is 



dxdydz r':i .cdt,{l'-£TA^^*) . . . (d); 



and the part of the heat taken from this element, which is rejected 

 into the refrigerator, is 



dx dy dz J cdt .e^J^ 

 — i~?T~7; ' 



dx dy dz^J"\dt ^f^'-f^cdt ir/y*^^ 



or 1 /.-p 



(«), 



These expressions are of course equally applicable to parts of 

 the body of which the temperatures are lower than T; and, 

 without change of form, will express respectively (in virtue of the 

 algebraic signs corresponding to any case in which ^ is < T), 

 {d) the quantity of work obtained by raising the temperature of an 

 element dx dy dz from ^ to T by heat drawn from a source at the 

 temperature T by means of a thermo-dynamic engine, and {e) the 



